SMALL POINTS AND ADELIC METRICS Shouwu Zhang Department of Mathematics Princeton University Princeton, NJ 08540 szhang@math.princeton.edu April 1993, Revised April 1994

SMALL POINTS AND ADELIC METRICS

Shouwu Zhang

Department of Mathematics Princeton University Princeton, NJ 08540 szhang@math.princeton.edu April 1993, Revised April 1994

Introduction

Consider following generalized Bogomolov conjecture: LetA be an abelian variety over Q,¯ h:A( ¯Q)→R a N´eron - Tate height function with respect to an ample and symmetric line bundle on A, Y a subvariety of A which is not a translate of an abelian subvariety by a torsion point; then there is a positive number ² such that the set

Y_² ={x∈Y( ¯Q) :h(x)≤²}

is not Zariski dense in Y. Replacing A by an abelian subvariety, we may assume that Y −Y = {y1 −y2 : y1, y2 ∈ Y( ¯Q)} generates A: A is the only abelian subvariety of A which contains Y −Y.

In this paper, we will prove thatY² is not Zariski dense if the map NS(A)R →NS(Y)R

is not injective.

In spirit of Szpiro’s paper [Sz], we will reduce the problem to the positivity of the height of Y with respect to certain metrized line bundle. To do this, we will first extend Gillet- Soul´e’s intersection theory of hermitian line bundles to certain limits of line bundles which are called integrable metrized line bundles, then for a dynamic system, construct certain special integrable metrized line bundles which are called admissible metrized line bundles, and finally prove the positivity of heights.

Integrable metrized line bundles. Consider a projective varietyX over Spec Q. For a line bundleL onX and an arithmetic model (X,e L) of (X,e L^e) over Spec Z, one can define an adelic metric k · kL˜=©

k · kp, p∈ Sª

on L, wheree is a positive integer,S is the set of places of Q, and k · kp is a metric on L ⊗QQ_p on X(Q_p).

Let L1,· · ·,Ld (d = dimX+ 1) be line bundles on X. For each positive integer n, let (Xen,Le1,n,· · · ,Led,n) be an arithmetic model of (X,L^e₁^1,n,· · · L^e_d^d,n) on Spec Z. Assume for each i that (L,k · kLei,n) converges to an adelic metrized line bundle Li. One might ask whether the number

cn = c1(Le1,n)· · ·c1(Led,n) e1,n · · · ed,n

in Gillet-Soul´e’s intersection theory converges or not.

We will show thatcnconverges if allLei,n are relatively semipositive, and that limn→∞cn

depends only on Li. Notice that some special case has been studied by Chinberg, Rumely, and Lau [CRL]. We say that an adelic line bundleL is integrable if L= ¯^∼ L1⊗L¯⁻¹₂ with ¯Li

semipositive. It follows that Gillet-Soul´e’s theory can be extended to integrable metrized line bundles. Some theorems such as Hilbert-Samuel formula, Nakai-Moishezon theorem, and comparison inequality remain valid on integrable metrized line bundles.

Admissible metrized line bundles. Letf :X →X be a surjective endomorphism over Spec Q, La line bundle on X, and φ:L^d 'f^∗L an isomorphism withd >1. Using Tate’s argument, we will construct a unique integrable metric k · k onL such that

k · k^d =φ^∗f^∗k · k.

If X =A is an abelian variety, and s is a section of L ⊗_QQ_p, then logksk_p is the N´eron function for divisor div (s).

In case thatL is ample, any effective cycle Y of X of pure dimension has an (absolute) height

h_L(Y) = c₁(L¯

¯Y)^dimY+1 (dimY + 1) deg_L(Y) which has property that

hL

¡f(Y)¢

=dhL(Y).

As Tate did,hL can be defined without admissible metric. Some situations are studied by Philippon [P], Kramer [K], Call and Silverman [CS], and Gubler [G].

Assume L is ample as above. If Y is preperiodic: the orbit {Y, f(Y), f²(Y),· · · } is finite, thenhL(Y) = 0. We propose a generalized Bogomolov conjecture which claims that the converse is true: if h(Y) = 0 then Y is preperiodic. This is a theorem [Z2] for case of multiplicative group. A consequence is the generalized Lang’s conjecture which claims that if Y is not preperiodic then the set of preperiodic points in Y is not Zariski dense.

Lang’s conjecture is proved by Laurent [L] and Sarnak [Sa] for multiplicative groups, and by Raynaud [R] for abelian varieties.

Positivity of heights of certain subvarieties. Let Y be a subvariety of an abelian variety A with a polarization L. We prove the following special case of the generalized Bogomolov conjecture: if Y − Y generates A, and the map NS(A)Q → NS(Y)Q is not injective, then hL(Y)> 0. The crucial facts used in the proof are comparison theorem of heights, Faltings index theorem, and nonvanishing of invariant (1,1) forms on Y.

For a curve C of genus g ≥ 2, let ω denote the admissible dualizing sheaf defined in [Z1]. The above positivity implies that ω² > 0 if End ¡

Jac(C)¢

R is not isomorphic to R,C, and the quaternion divison algebra D. This is the case when Jac(C) has a complex multiplication, or C has a finite morphism of deg>1 to a nonrational curve. Notice that if C has good reduction everywhere, and Jac(C) has complex multiplication, the positivity of ω²_Ar is proved by Burnol [B] using Weierstrass points.

1. Integrable meterized line bundles

(1.1). For a line bundleL on a projective schemeX over an algebraically closed valuation field K, we define a K-metric k · k on L to be a collection of K-norms on each fiber L(x), x∈X(K).

For example when K is non-archimedean, if there is a projective scheme Xe on SpecR with generic fiber X, and a line bundleLeon Xe whose restriction onX isL^⊗n, where Ris the valuation ring ofK and n >0 is an integer, we can define a metric k · kL˜ as follows:

For an algebraic pointx∈X(K), denote by e

x: Spec R−→Xe the section extending x: x = xe¯

¯SpecK, then xe^∗L ⊗R K = x^∗Lⁿ. For any ` ∈ x^∗(L), we define

k`kL˜ = inf

a∈K

©|a|ⁿ¹ : ` ∈aex^∗(L)ª . We say that k · kL˜ is induced by the model (X,e L).e

A metric k · k on L is called continuous and bounded if there is a model (X,e L) suche that log_k·k^k·k

L˜ is bounded and continuous on X(K) with respect to the K-topology.

(1.2). Denote by S ={∞,2,3,· · · } the set of all places of Q. For each p∈ S, denote by

| · |p the valuation onQsuch that|p|p =p⁻¹ ifp6=∞, by | · |∞ the ordinary absolute value if p=∞, byQ_p the completion ofQunder| · |_p, and by ¯Q_p a fixed algebraic closure ofQ_p. For an irreducible projective variety X over Q, we define an adelic metrized line bundle Le to be a line bundle L on X and a collection of metrics k · k ={k · kp, p∈ S} such that the following conditions are verified.

(a) Eachk · k_p is bounded, continuous, and Gal ¡Q¯_p/Q_p¢

invariant.

(b) There is a Zariski open subset U = Spec Z£₁

n

¤ of Spec Z, a projective variety Xe on U with generic fiber X, and a line bundle Leon Xe extending L on X, such that for each p∈U, the metric k · kp is induced by the model

¡Xep,Lep

¢=¡Xe ×U Spec ¯Zp,L ⊗e _Z[n¹]Z¯p

¢, where ¯Z_p denotes the valuation ring of ¯Q_p.

For example, if there is a projective variety Xe on Spec Z with generic fiber X, and a hermitian line bundleLeon Xe whose restriction on X is L^⊗n (n6= 0), then (X,e L) inducese a metric k · k_Le = ©

k · k_p, p ∈ Sª

, where for p 6= ∞,k · k_p is induced by models ¡Xe_p,Le_p¢ , and k · k∞ is the hermitian metric on LC. The condition (a) is obviously verified. Since L is defined over the generic fiber of X, there is an open subsete U⁰ of Spec Z such that L has an extention Le₁ on Xe_U⁰. Since Leⁿ₁¯

¯X = Le¯

¯X, there is an open subset U of U⁰ such that Leⁿ₁¯¯

U 'Le¯¯

U. It follows that for p∈ U , k · kp is induced by Le1. The condition (b) is therefore verified.

A sequnce{k · kn: n= 1,2,· · · } of adelic metrics is convergent to an adelic metrick · k if there is an open subset U of Spec Z such that for each p∈ U, k · k_n,p =k · k_p for alln, and that log^k·k_k·k^n,p converges to 0 uniformly on X(K).

(1.3). For a hermitian line bundle Leon an arithmetic variety X with a smooth metric at

∞, this means that for any holomorphic map

f :D={z ∈C:|z|<1} →X(C)

the pullback metric on f^∗L is smooth, we say that Le is relatively semipositive if Le has nonnegative degree on any curve in special fibers, and the curvature of f^∗LC is semipos- itive for any holomorphic map f : D → X(C); we say that Le is relatively ample if the associated algebraic bundle is relatively ample, and there is an embeddingX(C)e →Y into a projective complex manifoldY such that the hermitian line bundleL(C) can be extendede to a hermitian line bundle on Y with positive curvature.

For a projective varietyX over Spec Q, and a line bundleLonX, we say that an adelic metrick · k onL is ample (resp. semipositive) if it is the limit of a sequencek · kn of adelic metrics induced by models ¡Xe_n,Le_n¢

as in (1.2) such that Le_n are relatively ample (resp.

relatively semipositive.)

Let ¯L₁,· · · ,L¯_d be metrized line bundles onX with semipositive metrics,d= dimX+ 1.

Assume that k · ki are approximated by metrics induced by models¡Xei,n,Lei,n

¢, where Lei,n

are semipositive such that Lei,n

¯¯

X = L^ei,n, with ei,n > 0. For any d-tuple of positive integers (n1,· · ·, nd), denote byXen1,···,nd the Zariski closure of ∆(X) in Xen1 ×Z Xen2 ×Z

· · · ×ZXend, where ∆ is the diagonal map ofX into the generic fiber X ×QX × · · · ×QX of Xen1 ×ZXen2 ×Z· · · ×ZXend. We still denote the pullback ofLei,ni on Xen1,···,nd by Lei,ni. Theorem (1.4). (a) The intersection number

cn1,···,nd =c1(Len1)· · ·c1(Lend)±

e1,n1· · ·ed,nd

converges as ni → ∞. The limit does not depend on the choice of (Xei,n,Lei,n).

(b) Denoted byc1( ¯L1)· · ·c1( ¯Ld)the limit, thenc1( ¯L1)· · ·( ¯Ld)is multi-linear inL¯1,· · ·,L¯d. Proof. (a) Fix two d-tuples (n₁,· · · , n_d) and (n⁰₁,· · · , n⁰_d) of positive integers. Denote by Xe the Zariski closure of ∆(X) in Xen1,···,nd×Xen⁰₁,···,n⁰_d. As before we use same notations for pullbacks of Lei,ni as themselves. Write Lei = Le^e_i,n^i,n_i⁰ⁱ,Le⁰_i = Le^e_i,n^i,ni_i , and ei = ei,ni ·ei,n⁰_i. Then both Le_i and Le⁰_i have same restriction L^ei onX. We need to show that

1 e1· · ·ed

¡c₁(Le₁)· · ·c₁(Le_d)−c₁(Le⁰₁)· · ·c₁(Le⁰_d)¢

approaches to 0 as (n₁,· · · , n₂, n⁰₁,· · · , n⁰_d) approaches to ∞.

Fix a positive number ² and an open subset U of Spec Z such that for any p ∈U and any k, k · k_p,Le

k =k · k_p,Le0

k. Then for any sufficiently large n_k, n⁰_k and any p,

¯¯

¯¯log k · k_p,Le⁰_k

k · k_p,Lek

¯¯

¯¯≤²logp,

where log∞ is defined to be 1. Denote by sk the rational section of Lek ⊗Le⁰⁽⁻¹⁾_k which gives 1 on X.

Ifp6=∞, then one has

p^{−²e1···ed} ≤ kskkp(x)≤p^²e1···ed for any x. Assume [div (sk)]p = P

ni,pVi,p, where [div (sk)]p is the cycle associated to div (sk) supported in the special fiberXepofXe overp, andVi,p’s are irreducible components of Xe_p. It follows that |n_p| ≤²e₁· · ·e_d, or in other words, that divisors

D1p = [div (sk)]p+ [²ei· · ·en][Xep] and

D_2p =−[div (s_k)]_p+ [²e_i· · ·e_d][Xe_p]

are both effective, where [²e1· · ·ed] is the integral part of ²ei· · ·ed. Therefore fori= 1,2, c₁¡Le⁰¯

¯Di,p

¢· · ·c₁¡Le⁰_k−1¯

¯Di,p

¢c₁¡Le_k+1¯

¯Di,p

¢· · ·c₁¡Le_d¯

¯Di,p

¢>0,

or in other words, the contribution at p of

Ik =c1(Le⁰)· · ·c1(Le⁰_k−1)c1(Lek+1)· · ·c1(Led)div (s) has absolute value bounded by

²e1· · ·ed(logp)c1(L1)· · ·c1(Lk−1)c1(Lk+1)· · ·c1(Ld).

Ifp=∞, the contribution at p of Ik is given by Z

logks_kk_∞c⁰₁(Le⁰₁)· · ·c⁰₁(Le⁰_k−1)c⁰₁(Le_k+1)· · ·c⁰₁(Le_d)

where c⁰₁(Lei) and c⁰₁(Le⁰_i) denote the curvatures of Lei and Le⁰_i. Since |logkskk∞

¯¯ < ² and c⁰₁(Lei), c⁰_i(Le⁰_i) are nonnegative, the above integral has absolute value bounded by

²e1· · ·edc1(L1)· · ·c1(Lk−1)c1(Lk+1)· · ·c1(Ld) It follows that for any 1≤k ≤d,

|Ik| ≤²e1· · ·edc1(L1)· · ·c1(Lk−1)c1(Lk+1)· · ·c1(Ld)X

p /∈U

logp.

Finally, for (ni,· · · , nd, n⁰_i,· · · , n⁰_d) sufficiently large, 1

e1· · ·ed

¯¯

¯¯c1(Le1)· · ·c1(Led)−c1(Le⁰₁)· · ·c1(Le⁰_d)¯¯

≤ 1

e1· · ·ed

Xd

k=1

¯¯

¯¯c1(Le⁰₁)· · ·c1(Lek−1)c1(Lek⊗Le⁰_k⁻¹)c1(Lk+1)· · ·c1(Ld)¯¯

≤²· Xd

k=1

c₁(L₁)· · ·c₁(L_d)X

p /∈U

logp.

This prove the first statement of (a).

If©

(Xe_i,n⁰ ,Le⁰_i,n)ª

is another sequence of models which induces metricsk·k_Le

i,n0 convergent to k · k, then the alternating sequence

{(Xe_i,n⁰⁰ ,Le⁰⁰_i,n)}={¡Xe_i,1,Le_i,1¢

,¡Xe_i,1⁰ ,Le⁰_i,1¢

,¡Xe_i,2,Le_i,2),¡Xe_i,2⁰ ,Le⁰_i,2),· · · }

also induces metrics onL convergent tok · k. By the first statement of (a), the intersection numbers induced by {(Xe_i,n⁰⁰ ,Le⁰⁰_i,n)} are convergent. So limits defined by {(ex_i,n,Le_i,n)} and {ex⁰_i,n,Le⁰_i,n)} are the same. This prove the second statement of (a).

The additivity ofc₁( ¯L₁)· · ·c₁( ¯L_d) in (b) is obvious from definition.

This completes the proof of the theorem.

(1.5). A metrized line bundle ¯L = (L,k·k) is called integrable if there are two semipositive metrized line bundles ¯L1,L¯2 such that ¯L is isometric to ¯L1⊗L¯⁻¹₂ .

By theorem (1.4), for any integrable metrized line bundles ¯L1,· · · ,L¯d, there is a uniquely defined intersection numberc1( ¯L1)· · ·c1( ¯Ld) such that the following conditions are verified:

(a)c1( ¯L1)· · ·c1( ¯Ld) is multilinear

(b)c1( ¯L1)· · ·c1( ¯Ld) is the limit defined in (b) of (1.4) if the metrics on ¯L1,· · · ,L¯d are semipositive.

(1.6). Now we want to generalize results in [Z2] to integrable metrized line bundles. First of all, we need to define Hilbert function of a line bundle.

By a norm k · k on a vector space V of finite dimension over Q, we mean a collection {k · kp, p∈ S} of norms such that the following conditions are verified.

(a) For eachp,k · kp is a Qp-norm onVp =V ⊗QQp, which is nonarchimedean ifp6=∞, i.e. kx+ykp ≤max(kxkp,kykp) if p6=∞,

(b) There is a non zero integer n, a free module Ve over Z£₁

n

¤, and an isomorphism V 'Ve ⊗_Z[¹

n]Q such that k · kp is induced byVe for all p6 |n.

Denote by A the ring of adeles of Q, and by VA the module V ⊗QA. There is unique invariant measureµ on VA such thatµ¡ Q

p

Bp

¢= 1 where for eachp, Bp is the unit ball in V_p :B_p =©

x∈V_p,kxk_p ≤1ª

. We define the Euler characteristic of (V,k · k) as follows:

χ_k·k(V) =−log volume(VA

±V).

For a projective varietyX over Q, an ample line bundleL onX with a semipositive metric k · k, and a place pofQ, let k · kp denote a norm on Γ(L)⊗QQp = Γ(Lp) defined as follows:

for each `∈Γ(Xp,Lp),

k`k_p = sup

x∈X(Q_p)

k`k(x).

In this way, k · k=©

k · k_p, p∈ Sª

defines an adelic norm on Γ(L). Write χ¡ Γ( ¯L)¢

simply for χk·k

¡Γ( ¯L)¢ .

Theorem (1.7). As napproaches ∞, χ¡

Γ(L^⊗n)¢

= n^d

d!c1( ¯L)^d +o(n^d).

Proof. Assume that there is a sequence of models (Xe_m,Le_m) of (X,L) such that Le_m’s are semipositive on Xem and that the induced adelic metrized line bundles ¯Lm converge to ¯L, then the theorem is true for ¯Lm by (1.4) of [Z2]. Write

χ_m,n = χ¡

Γ( ¯L^⊗n_m )¢ ndim Γ(L^⊗n), χn = χ¡

Γ( ¯L^⊗n)¢ ndim Γ(L^⊗n),

then χn,m −→ χn uniformly in n as m → ∞. Now the theorem for ¯Lm implies that

n→∞lim χm,n= _dc^{c1( ¯Lm)d}

1(L)^d−1. It follows that

n→∞lim χn = lim

m→∞ lim

n→∞χm,n= c1( ¯L)^d dc1(L)^d−1. The theorem follows immediately.

Theorem (1.8). LetL¯be an ample metrized line bundle with an ample metric. Assume for each irreducible subvariety Y of X that c1( ¯L|Y)^dimY+1 > 0. Then for n À 0, the Q-vector space Γ(L^⊗n) has a basis {`<sub>1</sub>,· · · , `_N} consisting of strictly effective elements:

k`ikp ≤1 for p6=∞ and k`ik∞ <1.

Proof. By (1.7) and the Minkowski theorem, for each Y of X there is a n > 0, such that Γ(L¯

¯^⊗n

Y ) has a section ` such that k`k_p ≤ 1 for all p 6= ∞ and k`k_∞ < 1. The theorem follows from (4.2) of [Z2].

(1.9). For a projective variety X over Spec Q of dimension d − 1, and an integrable metrized ample line bundle ¯LonX, we define the height of X with respect to ¯Las follows:

hL¯ = c₁( ¯L)^d dc1(LQ)^d−1. For i= 1,2,· · ·d, define numbers

ei( ¯L) = sup

codY=i

x∈X−Yinf hL¯(x) where Y runs through the set of reduced subvarieties of X.

Theorem (1.10). If L¯ is an ample metrized line bundle then e₁( ¯L)≥hL¯(X)≥ e1( ¯L) +· · ·+ed( ¯L)

d

Proof. Assume ¯Lis approximated by metrized line bundles ¯L_nwhich are induced by models (Xen,Len). Then the theorem is true for ¯Ln by (5.2) of [Z2]. Since c1( ¯Ln)^d →c1( ¯L)^d and ei( ¯Ln)→ei( ¯L), the theorem (1.10) is true for ¯L.

Theorem (1.11). If L¯1· · ·L¯d are ample metrized line bundles, then

c1( ¯L1)· · ·c1( ¯Ld)≥ Xd

k=1

ed( ¯Lk)c1(L1)· · ·c1(Lk−1)c1(Lk+1)· · ·c1(Ld)

Proof. By a limit argument, we may assume that ¯L1,· · · ,L¯d are exactly metrized line bundles associated to relatively ample hermitian line bundles Le1,· · · ,Led on a model Xe of X. By scaling metric at ∞, we may assume that e_d(Le_i) = 0. We want to prove that c1(Le1)· · ·c1(Led) ≥ 0 by induction on d. It is obviously true for d = 1. Fix a ² > 0 and denote byLed(²) the metrized line bundle which hase^−²· k · kLed as metric at∞. NowLed(²) has height ≥² on X(Q). By (1.8), Led(²) is ample. In particular, some power Led(²)^m has an effective section `. Write Y = div (`), then

c1(Le1)· · ·c1(Led)

=c1(Le1)· · ·c1(Led(²))−²c1( ¯L1)· · ·c1( ¯Ld−1)

=1

mc1(Le1)· · ·c1(Led−1)(Y,−logk`k∞)−²c1( ¯L1)· · ·c1( ¯Ld−1)

=1

mc1(Le1|Y)· · ·c1(Led−1|Y) + 1 m

Z

X(C)

−logk`k∞c⁰₁(Le1)· · ·c⁰₁(Led−1)−²c1( ¯L1)· · ·c1( ¯Ld−1).

The first two terms in the last line are positive, since the first term is positive by induction, and the second term is positive by fact that k`k∞ <1 and c⁰₁(Lei)≥0. Letting ²→0, the theorem follows.

2. Admissible metrized line bundles

(2.1). LetK be an algebraically closed valuation field,X a projective variety over Spec K, L = (L,k · k) a line bundle on X with a continuous and bounded metric, f : X → X a surjective morphism, and φ : L^d ' f^∗L an isomorphism where d > 1 is an integer. We define k · kn on L inductively as follows:

k · k1 =k · k, k · kn =φ^∗f^∗k · k_n−1^1d .

Theorem (2.2). (a) The metrics k · kn on L converge uniformly to a metric k · k0 on L, this means that the function log^k·k_k·kⁿ

1 converges uniformly on X(K) to log^k·k_k·k⁰

1.

(b)k · k0 is the unique (continuous and bounded) metric on L satisfying the equation k · k0 =¡

φ^∗f^∗k · k0

¢¹

d.

(c) Ifφ changes to λφ with λ ∈K^∗, then k · k0 changes to |λ|^d−11 k · k0. Proof. (a) Denote by h the continuous function ¹_dφ^∗f^∗log^k·k_k·k²

1 on X. Then logk · k_n =¡1

dφ^∗f^∗¢_n−2

logk · k₂

=¡1

dφ^∗f^∗¢_n−2

(h+ logk · k1)

=¡1

dφ^∗f^∗¢_n−2

h+ logk · kn−1. Using induction on n, one has

logk · kn= logk · k1+

n−2X

k=0

¡1

dφ^∗f^∗¢_k

·h.

Since °

°¡₁

dφ^∗f^∗¢_k

·h°

°sup ≤ _d¹kkhksup, it follows that P^∞

k=1

¡₁

dφ^∗f^∗¢_k

· h is absolutely and uniformly convergent to a bounded and continuous functionh0. Let k · k0 =k · k1e^h0, then k · kn converges uniformly to k · k0.

(b) It is easy to see thatk · k0 is continuous, bounded, and satisfies the equation k · k0 =¡

φ^∗f^∗k · k0

¢¹

d.

If k · k⁰₀ be another continuous, bounded metric on L which satisfies the same equation, writingg = log ^k·k_k·k⁰0

0, then we haveg= ^φ∗_d^f∗g, sokgksup=kgksup

±d, org= 0. This shows that k · k0 =k · k⁰₀.

(c) Ifαk · k0 is the metric corresponding to λφ withα a function onX(K), then for any

`∈ L(x), x∈X,

αk`k0(x) = µ

αkλφ(`)k0

¡f(x)¢¶¹_d , so α = (α|λ|)^1d or α =|λ|^d−11 .

The proof of the theorem is complete.

(2.3). Now let everything be defined over Q: X is a projective variety over Spec Q, L an ample line bundle on X, f : X →X a surjective morphism over Q, and φ: L^d 'f^∗L an isomorphism of line bundles. This implies thatf is finite of degree d^dimX. We fix a model (X,e L) of (X,e L^e) on Spec Z with e >0, such that Leis relatively ample. This induces an adelic metric k · k onL.

There is an open subset U of Spec Z such that f and φ extend to an U-morphism fU :XU −→XU and an isomorphism φU :Le^⊗d_U −→f_U^∗LeU.

It follows for each p∈U that

k · k_p = (φ^∗f^∗k · k_p)^1d.

We define the morphism fe_n : Xe_n −→ Xe as the normalization of the composition of morphism

XU f_Uⁿ

−→XU ,→X.e

Denote by ¯Ln the metrized line bundles (L,k · kn) induced by model ¡Xen,fe_n^∗Le¢

. Then for any p ∈ U and any n, one has k · k_n,p = k · k_p. In general for any p ∈ S, k · k_n,p is defined as in (2.1) from k · kp. By theorem (2.2.), k · kn,p converges uniformly to a metric k · k0,p. So the adelic metric k · kn of ¯Ln converges to an adelic metric k · k0 on L. By (2.2), k · k0 doesn’t depend on the choice of (X,e L). Ife φ changes to λφ for λ ∈Γ(X,O_X^∗ ), then k · k₀ changes to k · k_0λ ={k · k_p|λ|p^d−11 }. Therefore, if we write ¯L₀ = (L,k · k₀) then ¯L^d−1₀ does not depend on the choice of φ. Since ¯Ln are all ample, ¯L0 is an ample metrized line bundle on X.

For any effective cycleY of X of pure dimension, write hf,L(Y) for hL¯0(Y).

Theorem (2.4). (a) Denote by f(Y) the push-forward of Y under f, then hf,L(f Y) = dhf,L(Y).

(b)hf,L(Y)≥0.

(c) If the orbit{Y, f(Y),· · · , fⁿ(Y),· · · } is finite then hf,L(Y) = 0.

(d) If y ∈ X(Q) is a point and hf,L(y) = 0 then the orbit {y, f(y),· · · , fⁿ(y),· · · } is finite.

Proof. (a)

hf,L(f(Y)) =c1

¡L¯0

¯¯

f(Y)

¢_dimY+1Á

(dimY + 1)c1

¡L¯

¯f(Y)

¢_dimY

=c1

¡f^∗L¯0

¯¯

Y

¢_dimY+1Á

(dimY + 1)c1(f^∗L)^dimY

=c1

¡L¯^d₀¯

¯Y

¢_dimY+1Á

(dimY + 1)c1

¡L^d¯

¯Y

¢_dimY

=dc₁¡L¯₀¯

¯Y

¢_dimY+1Á

(dimY + 1)c₁¡ L¯

¯Y

¢_dimY

=dhf,L(f(Y)).

(b) Applying (1.10) to Y, we have that hf,l(Y)≥ed(L¯

¯Y)·¡

dimY + 1¢

≥ed

¡L¢¡

dimY + 1¢ . But

ed(L) = inf

x hL(x) = inf

x hL

¡f(x)¢

=dinf

x hL(x) =ded(L).

Therefore ed(L) = 0

(c) The finiteness of the orbit {Y, f(Y),· · · , fⁿ(Y),· · · } implies the finiteness of the orbit

{h_f,L(Y), dh_f,L(Y),· · · , dⁿh_f,L(Y)· · · }.

So we must have hf,L(Y) = 0.

(d) If hf,L(y) = 0 then the orbit {y, f(y),· · · } has bounded degree [Q(y) : Q] and bounded height (= 0), so must be a finite set.

The proof of the theorem is complete.

Conjecture (2.5). If Y is an effective cycle of X of positive dimension and hf,L(Y) = 0 then the orbit of Y under f is finite.

This conjecture is a converse of (c) in (2.4). A subvarietyZ ofX is called a preperiodic subvariety if the orbit of Z underf is finite. A preperiodic subvariety Z contained inY is called maximal preperiodic if no other preperiodic subvariety of Y contains Z.

If hf,L(Y) = 0, by conjecture (2.5), Y is a preperiodic variety, of course a maximal preperiodic subvariety ofY. Ifh_f,L(Y)6= 0, by theorem (1.10), there is a Zariski open set U of Y such thathf,L onU(Q) has a positive lower bound, and any preperiodic subvariety Z of Y will be contained inX−U. This shows that (2.5) implies the following conjecture:

Conjecture (2.6). Any subvarietyY ofX contains at most finitely many maximal prepe- riodic subvarieties.

(2.7). Letf₀,· · · , f_n ben+1-homogeneous polynomial of degree ofd >1 inn+1 variables z0,· · · , zn such that the only common zero of f0,· · · , fn is 0. Then

f : (z0,· · · , zn)−→¡

f0(z0,· · · , zn),· · ·, fn(z0,· · · , zn)¢

defines a morphism Pⁿ−→Pⁿ. One has a unique homomorphism φ:O(d)'f^∗O(1) such that φ(fi) =f^∗(zi), where we consider zi as sections of O(1).

When fi = z_i^d, the preperiodic subvarieties of Pⁿ are Zariski closure of translates of subgroups by torsion points of Gⁿ_m = ©

(z₀· · ·z_n) =z₀· · ·z_n 6= 0ª

. In this case, (2.6) is a theorem of Laurent [L], Sarnak [Sa], while (2.5) is a theorem in [Z2].

(2.8). As in (2.1), let f : X −→ X be a surjective morphism, d a positive integer, and Pic(X)_f,d the subgroup of Pic(X) consisting of line bundles L such that L^⊗d ' f^∗L.

Assume that Pic(X)f,d contains an ample line bundle of X. Then any line bundle L in Pic(X)f,d can be written as L1 ⊗ L⁻¹₂ for two ample line bundles L1,L2 in Pic(X)f,d. By (2.3), there are ample metrized line bundles ¯L₁,L¯₂ whose generic fibers are L₁,L₂ and ¯L^⊗d_i ' f^∗L¯i. Now ¯L = ¯L1 ⊗L¯⁻¹₂ is an integrable metrized line bundle on X, and

L¯^⊗d ' f^∗L. By theorem (2.2), ¯¯ L^(d−1) does not depend on the choice of ¯L1,L¯2. Let Pic(x)_f,d denote the group of integrable metrized line bundles ¯L such that ¯L^⊗d ' f^∗L.¯ Then we have shown that Pic(X)_f,d is generated by ample metrized elements. We call elements in Pic(X)_f,d admissible metrized line bundles. The following theorem is useful in the next section.

Theorem (2.9). Let Y ,→ X be a subvariety of dimension n, and L ∈¯ Pic(X)f,d an ample metrized line bundle such that hf,L(Y) = 0, then

c1( ¯L1

¯¯

Y)· · ·c1( ¯Ln+1

¯¯

Y) = 0 for any L¯₁,· · · ,L¯_n+1 in Pic(X)_f,d, where n= dimY.

Proof. Since Pic(X)_f,d is generated by ample metrized line bundles, we may assume L¯1,· · · ,L¯n+1 are ample metrized. Since L is ample, there is a positive integer m, and an ample line bundle L0 such that L^m' L0⊗ L1⊗ · · · ⊗ Ln+1. Put a metric on L0 such that ¯L^m 'L¯0⊗L¯1⊗ · · · ⊗L¯n+1 then ¯L^⊗d₀ 'f^∗L¯0. By theorem (2.4)(b), ¯L,L¯0,· · · ,L¯n+1

are all semipositive. For any n+ 1 integers i1,· · · , in+1 between 0 and n+ 1, the number c1( ¯Li1)· · ·c1( ¯Lin+1) is nonnegative by (1.11). Since

0 =mⁿ⁺¹¡ c1( ¯L¯¯

Y)ⁿ⁺¹¢

=c1( ¯L^m¯¯

Y)ⁿ⁺¹

= X

0≤i1≤i2≤···≤in+1≤n+1

c1( ¯Li1)· · ·c1( ¯Lin+1),

we must have c1( ¯L1)· · ·c1( ¯Ln+1) = 0.

3. Positivity of heights of certain subvarieties of an abelian variety (3.1). Consider an abelian variety A over a number field K. For any integer n, let [n]

denote the endomorphism of Adefined as the multiplication ofn. Then for any symmetric line bundle L of A,L^⊗n2 ' [n]^∗L. If X = A, f = [n] for a n > 1, then hL = hf,L is the usual N´eron - Tate height function studied by Philippon [P], Kramer [K], and Gubler [G].

In this case, conjecture (2.5) is a theorem of Raynaud [R], and (2.6) is a conjecture of Bogomolov [B] if dimY = 1.

Theorem (3.2). Let Lbe a symmetric ample line bundle on A, and Y ,→A a subvariety of positive dimension such thatY −Y generates A. This means that A is the only abelian subvariety of A which containsY −Y. Assume that the induced map

NS(A)Q −→NS(Y)Q

is not injective, where NS(A) = Pic(A)±

Pic⁰(A) and NS(Y) = Pic(Y)±

Pic⁰(Y). Then hL(Y)>0.

(3.3). The crucial facts used in the proof of the theorem are theorem (2.9), a variant form (3.4) of Faltings’ index theorem [F1], and a nonvanishing theorem (3.5) for restriction on Y of an invariant 1 - 1 form of A(C).

Lemma (3.4). Let X be a variety over Q, and L¯and M¯ two integrable line bundles on X with smooth metrics at ∞. Assume that L¯ is semipositive and ω = c⁰( ¯L) is positive on a dense subset of the regular part Xr(C) of X(C), and that M is in Pic⁰(X). Then c1( ¯M)²c1( ¯L)^d−1 ≤0, and the equalityc1( ¯M)²c1( ¯L)^d−1 = 0implies that the metric onM¯ has curvature 0 on Xr(C).

Proof. Let f :X⁰ →X be a resolution of singularities. ReplacingX byX⁰, ¯Lbyf^∗L, and¯ M¯ by f^∗M¯ we may assume thatX is regular. Choose a metrick · k⁰_M on Msuch that its curvature is 0, let ϕ= log^k·k_k·k^0M_¯

M.

Fix a positive number². By approximation, there is a model (X,e L,e M) such thatf (a)Leis a relatively semipositive line bundle onXe whose restriction onX isL^e1, e1 >0, and whose metric at ∞is the e1-th power of the metric of ¯L;

(b)Mfis a line bundle on X, whose restriction one X is M^e2, e2 >0, and whose metric at ∞ is e2-th power of the metric of ¯M;

(c)c1( ¯M)²c1( ¯L)^d−1 ≤ _ed−1¹

1 e²₂c1(M)cf 1(L)e^d−1+².

Denote by Mf⁰ the metrized line bundle on Xe which has same finite part as Mfon X,e and which has metric k · k⁰_M. Then

(d)c1(M)f ² =c1(Mf⁰)²+¡

0,−ϕ^∂_πi^∂¯ϕ¢

as cycles on X.e We claim that

(e)c1(Mf⁰)²c1(L)e ^d−1 ≤0.

Fix a relatively ample line bundleLe⁰ on X, thene

n→∞lim n^1−dc1(Mf⁰)²c1(Leⁿ⊗Le⁰)^d−1 =c1(Mf⁰)²c1(L)e^d−1.

Replacing LebyLeⁿ⊗Le⁰ for n= 1,2,· · ·, we may assume that Leis relatively ample. Now, c₁(L)e ^d−1 is represented by _m¹(Z, g_Z), where Z is an integral subvariety of X with a regular generic fiber, m >0 an integer. Since Mf⁰ has curvature 0, one has

c1(Mf⁰)²c1(L)e^d−1 = 1

mc1(Mf⁰¯¯

Z)². Now c₁(Mf⁰¯

¯Z)² ≤0 by the Faltings-Hodge index theorem. The claim is proved.

Combining (a)-(e) we have that

c1( ¯M)²c1( ¯L)^d−1 ≤ − Z

X(C)

ϕ∂∂¯

πiϕω^d−1+².

Since ω^d−1 ≥0 and ω^d−1 >0 on a dense subset of X(C), by letting² →0 it follows that c1( ¯M)²c1( ¯L)^d−1 ≤ −

Z ϕ∂∂¯

πiϕω^d−1 ≤0, and that R

ϕ^∂_πi^∂¯ϕω^d−1 = 0 if and only if ϕis locally constant.

Lemma (3.5). Let A be a complex abelian variety, and Y ,→ A a subvariety such that {y1−y2

¯¯y1, y2 ∈Y(C)} generates A. This means that A is the only abelian subvariety of A which contains Y −Y. Ifω is a 1-1 form onA which is invariant under translation and ω¯

¯Y = 0, then ω = 0.

Proof. We write A = Cⁿ±

Λ and ω = P

a_ijdz_iΛd¯z_j. After a translation, we may assume that 0 is a smooth point ofY. Fix pointsy1,· · · , ymonY such that{y1,· · · , ym}generates A. One can find a complex curve C ,→Y, such that y_i ∈C and 0 is a regular point of C.

Fix any holomorphic map ϕ:D={z ∈C:|z|<1},→Cⁿ±

Λ such thatϕ(D),→C. Write ϕ(z) =¡

f₁(z),· · · , f_n(z)¢

, thenf₁⁰(z),· · · , f_n⁰(z) are linearly independent over C.

Ifω¯

¯Y = 0 thenϕ^∗ω = 0, it follows that Xaij∂fi

∂z

∂f¯j

∂z¯ = 0

for all z ∈ D. Comparing coefficients of power series in z and ¯z, since ^∂f_∂zⁱ are linearly independent in C, for any i we must have

X

j

aij

∂f¯_j

∂¯z = 0, or

X

j

¯ aij

∂f_j

∂z = 0.

It follows that aij = 0 for all i, j. So ω = 0. The proof of the lemma is complete.